Symplectic resolutions of Quiver varieties and character varieties

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arXiv:1602.00164v1 [math.AG] 30 Jan 2016

GWYN BELLAMY AND TRAVIS SCHEDLER Abstract. In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. Namely, we consider the question of when a quiver variety admits a projective symplectic resolution. A complete answer to this question is given. We also show that the smooth locus of a quiver variety coincides with the locus of θ-canonically stable points, generalizing a result of Le Bruyn. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT. In the final part of the article, we consider the G-character variety of a compact Riemann surface of genus g > 0, when G is SL(n, C) or GL(n, C). We show that these varieties admit symplectic singularities. When the genus g is greater than one, we show that the singularities are terminal and locally factorial. As a consequence, these character varieties do not admit symplectic resolutions.

Table of Contents 1.



Quiver varieties


Canonical Decompositions


Smooth v.s. stable points


The variety X(n, d)


Divisible non-isotropic imaginary roots


Namikawa’s Weyl group


Character varieties

1. Introduction Nakajima’s quiver varieties [34], [35], have become ubiquitous throughout representation theory. For instance, they play a key role in the categorification of representations of Kac-Moody Lie algebras, and the corresponding theory of canonical bases. They provide also ´etale-local models of singularities appearing in many important moduli spaces, together with, in most cases, a canonical symplectic resolution given by varying the stability parameter. Surprisingly, there seems to be no explicit criterion in the literature for when a quiver variety admits a projective symplectic resolution (often, in applications, suitable sufficient conditions for 2010 Mathematics Subject Classification. 16S80, 17B63. Key words and phrases. symplectic resolution, quiver variety, character variety, Poisson variety. 1

their existence are provided, but they do not appear always to be necessary). The purpose of this article is to give such an explicit criterion. Following arguments of Kaledin, Lehn and Sorger (who consider the surprisingly similar case of moduli spaces of semi-stable sheaves on a K3 or abelian surface), our classification result ultimately relies upon a deep result of Drezet on the local factorality of certain GIT quotients. Our classification begins by generalizing Crawley-Boevey’s decomposition theorem [7] of affine quiver varieties into products of such varieties (let us call them irreducible for now), to the nonaffine case (i.e., to quiver varieties with nonzero stability condition) (Theorem 1.3). Along the way, we also generalize Le Bruyn’s [29, Theorem 3.2], which computes the smooth locus of these varieties, again from the affine to nonaffine setting (Theorem 1.13). Then, our main result, Theorem 1.4, states that those quiver varieties admitting resolutions are exactly those whose irreducible factors, as above, are one of the following three types of varieties: (a) Varieties whose dimension vector are indivisible non-isotropic imaginary roots for the KacMoody Lie algebra associated to the quiver (so of dimension ≥ 4);

(b) Symmetric powers of deformations or partial resolutions of du Val singularities (C2 /Γ for Γ < SL2 (C)); (c) Varieties whose dimension vector are twice an indivisible non-isotropic imaginary root whose Cartan pairing with itself is −2. The last type is the most interesting one, and is closely related to O’Grady’s examples [30]: in this case, one cannot fully resolve or smoothly deform via a quiver variety, but after maximally smoothing in this way, the remaining singularities are ´etale-equivalent to the product of V = C4 with the locus of square-zero matrices in sp(V ) (which O’Grady considers, see also [30]), and the later locus is resolved by the cotangent bundle of the Lagrangian Grassmannian of V . In the case of type (a), one can resolve or deform by varying the quiver parameters, whereas in the case of type (b), one cannot resolve in this way, but the variety is well-known to be isomorphic to another quiver variety (whose quiver is obtained by adding an additional vertex, usually called a framing, and arrows from it to the other vertices), which does admit a resolution via varying the parameters. Moreover, in this case, if the stability parameter is chosen to lie in the appropriate chamber, then the resulting resolution is a punctual Hilbert scheme of the minimal resolution of the original (deformed) du Val singularity.

1.1. Symplectic resolutions. In order to state precisely our main results, we recall some standard notation. Let Q = (Q0 , Q1 ) be a quiver with finitely many vertices and arrows. We fix a dimension vector α ∈ NQ0 , deformation parameter λ ∈ CQ0 , and stability parameter θ ∈ QQ0 , such that

λ · α = θ · α = 0. Unless otherwise stated, we make the following assumption throughout the paper: If θ 6= 0 then λ ∈ RQ0 . 2


Associated to this data is the (generally singular) variety, which Nakajima defined and called a “quiver variety,” see Section 2 for details, Mλ (α, θ) := µ−1 (λ)θ //G(α). Remark 1.1. The construction in [34, 35] is apparently more general, depending on an additional dimension vector, called the framing. However, as observed by Crawley-Boevey [6], every framed variety can be identified with an unframed one. In more detail, for the variety as in [34, 35] with framing β ∈ NQ0 , it is observed in [6, §1] that the resulting variety can alternatively be f1 ), where Q f1 consists of Q1 together constructed by replacing Q by the new quiver (Q0 ∪ {∞}, Q

with, for every i ∈ Q0 , βi new arrows from ∞ to i; then Nakajima’s β-framed variety is the same P as M(λ,0) ((α, 1), (θ, i∈Q0 −θi )). Thus, for the purposes of the questions addressed in this article, it is sufficient to consider the unframed varieties.

+ + denote those positive roots of Q that pair to zero with both λ and θ. If α ∈ / NRλ,θ Let Rλ,θ

+ then Mλ (α, θ) = ∅, therefore we assume α ∈ NRλ,θ . Recall that a normal variety X is said to be

a symplectic singularity if there exists a (algebraic) symplectic 2-form ω on the smooth locus of X such that π ∗ ω extends to a regular 2-form on the whole of Y , for any resolution of singularities π : Y → X. We say that π is a symplectic resolution if π ∗ ω extends to a non-degenerate 2-form on Y.

Proposition 1.2. The variety Mλ (α, θ) is an irreducible symplectic singularity. From both the representation theoretic and the geometric point of view, it is important to know when the variety Mλ (α, θ) admits a symplectic resolution. In this article, we address this question, giving a complete answer. The first step is to reduce to the case where α is a root for which there exists a θ-stable representation of dimension α of the deformed preprojective algebra Πλ (Q). This is done via the canonical decomposition of α, as described by Crawley-Boevey; it is analogous to Kac’s canonical decomposition. Associated to λ, θ is a set Σλ,θ ⊂ R+ (which we will define in §2 below). Then α admits a canonical decomposition

α = n1 σ (1) + · · · + nk σ (k)


with σ (i) ∈ Σλ,θ pairwise distinct, such that any other decomposition of α into a sum of roots belonging to Σλ,θ is a refinement of the decomposition (2). Crawley-Boevey’s Decomposition Theorem [7], which we will show holds in somewhat greater generality, then says that: Theorem 1.3. The symplectic variety Mλ (α, θ) is isomorphic to S n1 Mλ (σ (1) , θ)×· · ·×S nk Mλ (σ (k) , θ) and Mλ (α, θ) admits a projective symplectic resolution if and only if each Mλ (σ (i) , θ) admits a projective symplectic resolution. Therefore it suffices to assume that α ∈ Σλ,θ . We write gcd(α) for the greatest common divisor

of the integers {αi }i∈Q0 . The dimension vector α is said to be divisible if gcd(α) > 1. Otherwise, it 3

is indivisible. Our main theorem states the following. Let p(α) := 1 − 12 (α, α) where (−, −) is the Cartan matrix associated to the undirected graph underlying the quiver, i.e., (ei , ej ) = 2 − |{a ∈ Q1 : a : i → j or a : j → i} for elementary vectors ei , ej .

+ Theorem 1.4. Let α ∈ NRλ,θ . The quiver variety Mλ (α, θ) admits a projective symplectic resolu-

tion if and only if each non-isotropic imaginary root σ appearing in the canonical decomposition of  α is either indivisible or gcd(σ), p gcd(σ)−1 σ = (2, 2). If α ∈ Σλ,θ is an indivisible non-isotropic imaginary root, then a projective symplectic resolution

of Mλ (α, θ) is given by moving θ to a generic stability parameter. However, this fails if α is a non isotropic imaginary root such that gcd(α), p gcd(α)−1 α = (2, 2). It seems unlikely that Mλ (α, θ) can be resolved by another quiver variety in this case. Instead, we show that the 10-dimensional

symplectic singularity Mλ (α, θ) can be resolved by blowing up a certain sheaf of ideals. Let θ ′ be a generic stability parameter such that θ ′ ≥ θ; see section 2.4 for the definition of ≥. fλ (α, θ ′ ) is the blowup of Theorem 1.5. There exists a sheaf of ideals I on Mλ (α, θ ′ ) such that if M fλ (α, θ ′ ) → Mλ (α, θ) is a projective symplectic Mλ (α, θ ′ ) along I, then the canonical morphism π : M resolution of singularities.

Set-theoretically, the set of zeros of I is precisely the singular locus of Mλ (α, θ ′ ). Remark 1.6. Since our reduction arguments are similar to those of [26], it is not so surprising (in hindsight at least) that Theorem 1.4 is completely analogous to [26, Theorems A & B]. In both cases, it is self-extensions of a certain kind that cannot be resolved symplectically. 1.2. Divisible non-isotropic imaginary roots. The real difficulty in the proof of Theorem 1.4 is in showing that if α ∈ Σλ,θ is a divisible non-isotropic imaginary root such that  gcd(α), p gcd(α)−1 α 6= (2, 2),

then Mλ (α, θ) does not admit a projective symplectic resolution. Based upon a deep result of Drezet [13], who considered instead the moduli space of semi-stable sheaves on a rational surface, we show in Corollary 6.8 that Theorem 1.7. Assume that θ is generic. The quiver variety Mλ (α, θ) is locally factorial. Since it is clear that Mλ (α, θ) has terminal singularities, the above theorem implies that it cannot admit a projective symplectic resolution. In fact, we prove a more precise statement than Theorem 1.7, see Corollary 6.8, which does not require that θ be generic. From Corollary 6.8, we deduce that Mλ (α, θ) does not admit a projective symplectic resolution. In fact, by the argument given in the proof of Theorem 6.13, we see that the corollary implies that this statement holds for open subsets of Mλ (α, θ): 4

Corollary 1.8. If U ⊆ Mλ (α, θ) is any singular Zariski open subset, then U does not admit a symplectic resolution.

In particular, for many choices of U , we obtain a variety which formally locally admits a symplectic resolution everywhere, but does not admit one globally. For example, if α = 2β for some β ∈ Σλ,θ with p(β) ≥ 3 (cf. the definition of p above Theorem 1.4), then we can let U be the

complement of the locus of representations X of the doubled quiver in µ−1 α (λ) which decompose as X = Y ⊕2 for Y a simple representation of dimension vector β. Remark 1.9. One does not need the full strength of the above theorem to show that Mλ (α, θ) does

not admit a symplectic resolution: it suffices to show that a formal neighborhood of some point does not admit a symplectic resolution. This can actually be deduced from a result of Kaledin, Lehn, and Sorger: see Remark 3.5 below for details. However, this does not actually simplify the proof since those authors also appeal to Drezet’s result in the same way (which is indeed where we learned about it). Moreover, this would not be enough to imply Corollary 1.8. There is one quiver variety in particular that captures the “unresolvable” singularities of Mλ (α, θ). This variety, which we denote X(n, d) with n, d ∈ N, has been extensively studied in the works of

Lehn, Kaledin and Sorger. Concretely, X(n, d) :=

d ) X [Xi , Yi ] = 0 // GL(n, C), (X1 , Y1 , . . . , Xd , Yd ) ∈ EndC (Cn )



Viewed as a special case of Corollary 6.8, it is shown in [26] that

Theorem 1.10. Let n, d ≥ 2, with (n, d) 6= (2, 2). The symplectic variety X(n, d) is locally factorial

and terminal. In particular, it has no projective symplectic resolution.

When d = 1, the Hilbert scheme of n points in the plane provides a symplectic resolution of X(n, d) ≃ S n C2 ; see [17, Theorem 1.2.1, Lemma 2.8.3]. When n = 1, one has X(n, d) ≃ A2d . Remark 1.11. It is interesting to note that [6, Theorem 1.1] implies that the moment map (X1 , Y1 , . . . , Xd , Yd ) 7→

d X [Xi , Yi ] i=1

is flat when d > 1, in contrast to the case d = 1, which is easily seen not to be flat. Remark 1.12. Generalizing the Geiseker moduli spaces that arise from framings of the Jordan quiver, it seems likely that the framed versions of X(n, d) (which are smooth for generic stability parameters) should have interesting combinatorial and representation theoretic properties. 5

1.3. Smooth versus canonically-stable points. In order to decide when the variety Mλ (α, θ) is smooth, we describe the smooth locus in terms of θ-stable representations. Write the canonical + decomposition n1 σ (1) + · · · + nk σ (k) of α ∈ NRλ,θ as τ (1) + · · · + τ (ℓ) , where a given root τ ∈ Σλ,θ

may appear multiple times. Recall that a point x ∈ Mα (λ, θ) is said to be polystable if it is a direct

sum of θ-stable representations. We say that x is θ-canonically stable if x = x1 ⊕ · · · ⊕ xℓ where

each xi is θ-stable, dim xi = τ (i) and xi 6≃ xj for i 6= j. The set of θ-canonically stable points in

Mλ (α, θ) is a dense open subset. When θ = 0, the result below is due to Le Bruyn [29, Theorem 3.2] (whose arguments we generalize). Theorem 1.13. A point x ∈ Mλ (α, θ) belongs to the smooth locus if and only if it is θ-canonically stable.

Remark 1.14. Theorem 1.13 confirms the expectation stated after Lemma 4.4 of [21]. An element σ ∈ Σλ,θ is said to be minimal if there are no β (1) , . . . , β (r) ∈ Σλ,θ , with r ≥ 2, such

that σ = β (1) + · · · + β (r) .

Corollary 1.15. The variety Mλ (α, θ) is smooth if, and only if, in the canonical decomposition α = n1 σ (1) + · · · + nk σ (k) of α, each σ (i) is minimal, and occurs with multiplicity one whenever σ (i)

is imaginary.

1.4. Namikawa’s Weyl group. When both λ and θ are zero, M0 (α, 0) is an affine conic symplectic singularity. Associated to M0 (α, 0) is Namikawa’s Weyl group W , a finite reflection group. In order to compute W , one needs to describe the codimension two symplectic leaves of M0 (α, 0). More generally, we consider the codimension two leaves in a general quiver variety Mλ (α, θ). We show that these are parameterized by isotropic decompositions of α. Definition 1.16. The decomposition α = β (1) + · · · + β (s) + m1 γ (1) + · · · mt γ (t) is said to be an isotropic decomposition if (1) β (i) , γ (j) ∈ Σλ,θ .

(2) The β (i) are pairwise distinct imaginary roots. (3) The γ (i) are pairwise distinct real roots. ′′

(4) If Q is the quiver with s+t vertices without loops and −(α(i) , α(j) ) arrows between vertices i 6= j, where α(i) , α(j) ∈ {β (1) , . . . , β (s) , γ (1) , . . . , γ (t) }, then Q′′ is an affine Dynkin quiver.

(5) The dimension vector (1, . . . , 1, m1 , . . . , mt ) of Q′′ (where there are s one’s) equals δ, the minimal imaginary root. Theorem 1.17. Let α ∈ Σλ,θ be imaginary. Then the codimension two strata of Mλ (α, θ) are in bijection with the isotropic decompositions of α.


1.5. Character varieties. The methods we use seem to be applicable to many other situations. Indeed, as we have noted previously, they were first developed by Kaledin-Lehn-Soerger in the context of semi-stable sheaves on a K3 or abelian surface. Any situation where the symplectic singularity is constructed as a Hamiltonian reduction with respect to a reductive group of type A is amenable to this sort of analysis. One such situation, which is of crucial importance in geometric group theory, is that of character varieties of a Riemannian surface. Let Σ be a compact Riemannian surface of genus g > 0 and π its fundamental group. The SL-character variety of Σ is the affine quotient Y(n, g) := Hom(π, SL(n, C))// SL(n, C). If g > 1 then dim Y(n, g) = 2(g − 1)(n2 − 1), and when g = 1, it has dimension 2(n − 1). We do

not consider the case where Σ has punctures, since the corresponding character variety is smooth in this case. Theorem 1.18. The variety Y(n, g) is an irreducible symplectic singularity.

The same arguments, using Drezet’s Theorem, that we have used to proof Theorem 1.7 are also applicable to the symplectic singularities Y(n, g). We show that: Theorem 1.19. Assume that g > 1 and (n, g) 6= (2, 2). The variety Y(n, g) has locally factorial, terminal singularities.

Arguing as in the proof of Theorem 6.13, Theorem 1.19 implies: Corollary 1.20. Assume g > 1 and (n, g) 6= (2, 2). Then the symplectic singularity Y(n, g) does

not admit a symplectic resolution. The same holds for any singular open subset.

Remark 1.21. Parallel to Remark 1.9, we can give an alternative proof of the first statement of Corollary 1.20 using formal localization, reducing to the quiver variety case. The formal neighborhood of the identity of Y(n, g) is well known to identify with the formal neighborhood of (0, . . . , 0) in the quotient (

d ) X [Xi , Yi ] = 0 // SL(n, C). (X1 , Y1 , . . . , Xd , Yd ) ∈ sl(n, C) i=1

This is (essentially) the formal neighborhood of zero of the quiver variety M(n) (0, 0) for the quiver Q with one vertex and g arrows. Since M(n) (0, 0) is conical, as we recall in Lemma 6.15 below, it admits a symplectic resolution if and only if its formal neighborhood of zero does. But the fact that this does not admit a resolution when g > 1 and (n, g) 6= (2, 2) is [26, Theorem B] (whose

proof we generalize to prove Theorem 1.19). However, we cannot directly conclude Theorem 1.19 using formal localization, and neither the stronger last statement of Corollary 1.20. 7

Remark 1.22. We expect that, as pointed out after Corollary 1.8, one can obtain singular open subsets U ⊆ Y(n, g) in the case g > 1 and (n, g) 6= (2, 2) for which the formal neighborhood of

every point does admit a resolution, even though the entire U does not admit one by the corollary.

Probably, one example is analogous to the one given there: for n = 2 and g ≥ 3, and U the

complement of the locus of representations of the form Y ⊕2 for Y one-dimensional (and hence irreducible). This would be such an example if Question 8.8 has a positive answer in the case V = Y ⊕2 (with n = 2 and g ≥ 3). Once again, the case of a genus two Riemann surface and 2-dimensional representations of π i.e. (n, g) = (2, 2), is special. In this case Y(2, 2) does not have terminal singularities. Moreover, by work of Lehn and Sorger [30], Y(2, 2) does admit a symplectic resolution. In fact an explicit resolution can be constructed. e 2) → Y(2, 2) of Y(2, 2) along the reduced ideal defining the Theorem 1.23. The blowup σ : Y(2, singular locus of Y(2, 2) defines a symplectic resolution of singularities.

Remark 1.24. When g = 1, the barycentric Hilbert scheme Hilbn0 (C× × C× ) provides a resolution

of singularities for Y(n, g).

In the body of the article, we consider instead the GL-character variety X(n, g) = Hom(π, GL(n, C))// GL(n, C). In section 8.6, we deduce Theorems 1.18, 1.19 and 1.23, and Corollary 1.20, from the corresponding results for X(n, g). Similar techniques are applicable to the Hitchin’s moduli spaces of semi-stable Higgs bundles over smooth projective curves. Details will appear in future work.

1.6. Acknowledgments. The first author was partially supported by EPSRC grant EP/N005058/1. The second author was partially supported by NSF Grant DMS-1406553. The authors are grateful to the University of Glasgow for the hospitality provided during the workshop “Symplectic representation theory”, where part of this work was done, and the second author to the 2015 Park City Mathematics Institute for an excellent working environment. We would like to thank Victor Ginzburg for suggesting we consider character varieties. We would also like to thank David Jordan, Johan Martens and Ben Martin for their many explanations regarding character varieties.

1.7. Proof of the main results. The proof of the theorems and corollaries stated in the introduction can be found in the following subsections. 8

Proposition 1.2 : Section 6.3 Theorem 1.3

: Section 6.4

Theorem 1.4

: Section 6.4

Theorem 1.5

: Section 5

Theorem 1.7

: Section 6.2

Corollary 1.8

: Section 6.4

Theorem 1.10

: –

Theorem 1.13

: Section 4

Corollary 1.15

: Section 4

Theorem 1.17

: Section 7

Theorem 1.18

: Section 8.6

Theorem 1.19

: Section 8.6

Corollary 1.20

: –

Theorem 1.23

: Section 8.6

Throughout, variety will mean a reduced, quasi-projective scheme of finite type over C. By symplectic manifold, we mean a smooth variety equipped with a non-degenerate closed algebraic 2-form. 2. Quiver varieties In this section we fix notation. 2.1. Notation. Let N := Z≥0 . We work over C throughout. All quivers considered will have a finite number of vertices and arrows. We allow Q to have loops at vertices. Let Q = (Q0 , Q1 ) be a quiver, where Q0 denotes the set of vertices and Q1 denotes the set of arrows. For a dimension vector α ∈ NQ0 , Rep(Q, α) will be the space of representations of Q of dimension α. The group Q G(α) := i∈Q0 GLαi (C) acts on Rep(Q, α); write g(α) = Lie G(α). The torus C× in G(α) of

diagonal matrices acts trivially on Rep(Q, α). Thus, the action factors through P G(α) := G(α)/C× . Let Q be the doubled quiver so that there is a natural identification T ∗ Rep(Q, α) = Rep(Q, α).

The group G(α) acts symplectically on Rep(Q, α) and the corresponding moment map is µ : Rep(Q, α) → g(α), where we have identified g(α) with its dual using the trace form. An ele-

ment λ ∈ CQ0 is identified with the tuple of scalar matrices (λi IdVi )i∈Q0 ∈ g(α). The affine quotient µ−1 (λ)//G(α) parameterizes semi-simple representations of the deformed preprojective algebra Πλ (Q); see [6] for details. If M is a finite dimensional Πλ (Q)-module, then dim M will always denote the dimension vector of M , and not just its total dimension. 2.2. Root systems. The coordinate vector at vertex i is denoted ei . The set NQ0 of dimension vectors is partially ordered by α ≥ β if αi ≥ βi for all i and we say that α > β if α ≥ β with α 6= β. 9

Following [8, Section 8], the vector α is called sincere if αi > 0 for all i. The Ringel form on ZQ0 is defined by hα, βi =

X i∈I

αi βi −


αt(a) βh(a) .


Let (α, β) = hα, βi + hβ, αi denote the corresponding Euler form and set p(α) = 1 − hα, αi. The

fundamental region F(Q) is the set of 0 6= α ∈ NQ0 with connected support and with (α, ei ) ≤ 0

for all i.

If i is a loopfree vertex, so p(ei ) = 0, there is a reflection si : ZQ0 → ZQ0 defined by si (α) =

α − (α, ei )ei . The real roots (respectively imaginary roots) are the elements of ZQ0 which can

be obtained from the coordinate vector at a loopfree vertex (respectively ± an element of the

fundamental region) by applying some sequence of reflections at loopfree vertices. Let R+ denote the set of positive roots. Recall that a root β is isotropic imaginary if p(β) = 1 and non-isotropic

imaginary if p(β) > 1. We say that a dimension vector α is indivisible if the greatest common divisor of the αi is one. 2.3. The canonical decomposition. In this section we recall the canonical decomposition defined by Crawley-Boevey (not to be confused with Kac’s canonical decomposition). Fix λ ∈ CQ0 and + θ ∈ QQ0 . Then Rλ,θ := {α ∈ R+ | λ · α = θ · α = 0}. Following [6], we define ( r   X + p β (i) for any decomposition Σλ,θ = α ∈ Rλ,θ p(α) > i=1

o + . α = β (1) + · · · + β (r) with r ≥ 2, β (i) ∈ Rλ,θ

+ Choosing a parameter λ′ ∈ CQ0 such that Rλ,θ = Rλ+′ , [7, Theorem 1.1] implies that1

+ . Then α admits a unique decomposition α = n1 σ (1) + · · · + nk σ (k) Proposition 2.1. Let α ∈ NRλ,θ

as a sum of element σ (i) ∈ Σλ,θ such that any other decomposition of α as a sum of elements from

Σλ,θ is a refinement of this decomposition.

The following elementary fact will be used frequently. Lemma 2.2. Let α be a non-isotropic imaginary root and m ∈ N. Then mα ∈ Σλ,θ if and only if

α ∈ Σλ,θ .

Proof. Assume that mα ∈ Σλ,θ . If α ∈ / Σλ,θ then α = β (1) + · · · + β (r) with r ≥ 2 and p(α) ≤ P (i) i p(β ). We have X X p(mα) = 1 − hmα, mαi = mp(α) − (m − 1) ≤ mp(β (i) ) − (m − 1) ≤ mp(β (i) ) i


which implies that mα ∈ / Σλ,θ . Conversely, if α ∈ Σλ,θ then [7, Proposition 1.2 (3)] says that

mα ∈ Σλ,θ .

1We don’t have to choose such a λ′ , since the arguments of [7] can be simply generalized to the context of the pair

(θ, λ). 10

2.4. Stability. Let θ ∈ QQ0 be a rational stability condition. Recall that a Πλ (Q)-representation

M , such that θ(M ) = 0, is said to be θ-stable, respectively θ-semi-stable, if θ(M ′ ) < 0, respectively

θ(M ′ ) ≤ 0, for all proper non-zero subrepresentations M ′ of M . A representation M is said to

be θ-polystable if M = M1 ⊕ · · · ⊕ Mk with θ(Mi ) = 0, such that each Mi is θ-stable. The set of θ-semi-stable points in µ−1 (λ) is denoted µ−1 (λ)θ . We define a partial order on QQ0 by setting

θ ′ ≥ θ if M θ-semistable implies that M is θ ′ -semi stable, i.e., θ′ ≥ θ


µ−1 (λ)θ ⊂ µ−1 (λ)θ .

The space Rep(Q, α) has a natural Poisson structure. Since the action of G(α) on Rep(Q, α) is Hamiltonian, Mλ (α, θ) = µ−1 (λ)θ //G(α) := Proj

M k≥0

is a Poisson variety.

ikθ h C µ−1 (λ)θ

Lemma 2.3. If θ ′ ≥ θ, then there is a projective Poisson morphism Mλ (α, θ ′ ) → Mλ (α, θ). ′

Proof. By definition, we have a G(α)-equivariant embedding µ−1 (λ)θ ֒→ µ−1 (λ)θ . This induces a morphism ′

Mλ (α, θ ′ ) = µ−1 (λ)θ //G(α) −→ µ−1 (λ)θ //G(α) = Mλ (α, θ), between geometric quotients. We need to show that this morphism is projective. This is local on Mλ (α, θ). Therefore we may choose n ≫ 0 and a nθ-semi-invariant f and consider the open subsets  ′ U ∩ µ−1 (λ)θ and U ∩ µ−1 (λ)θ , where U = (f 6= 0) ⊂ Rep(Q, α). Then U ∩ µ−1 (λ)θ //G(α) =  G(α) Spec C U ∩ µ−1 (λ)θ is an open subset of Mλ (α, θ) and ikθ′   M h ′ C U ∩ µ−1 (λ)θ U ∩ µ−1 (λ)θ //G(α) = Proj k≥0

   ′ such that U ∩ µ−1 (λ)θ //G(α) → U ∩ µ−1 (λ)θ //G(α) is the projective morphism Proj

M k≥0

iG(α) h ikθ′ h −→ Spec C U ∩ µ−1 (λ)θ . C U ∩ µ−1 (λ)θ

It is clear that this morphism is Poisson.

It follows from the proof of Lemma 2.3 that if θ ′′ ≥ θ ′ ≥ θ then the projective morphism

Mλ (α, θ ′′ ) → Mλ (α, θ) factors through Mλ (α, θ ′ ).

2.5. A stratification. Let x ∈ Mλ (α, θ) be a geometric point. We denote by the same symbol a point in the unique closed G(α)-orbit in µ−1 (λ)θ that maps to x. Then x decomposes into a direct

sum xe11 ⊕ · · · ⊕ xekk of θ-stable representations, with multiplicity. Let β (i) = dim xi . The point x

is said to have representation type τ = (e1 , β (1) ; . . . ; ek , β (k) ). Associated to x ∈ Mλ (α, θ)τ is the

stabilizer Gτ in G(α) of the lift of x in µ−1 (λ)θ . Even though µ−1 (λ)θ is not generally affine, the

fact that a non-zero morphism between θ-stable representations is an isomorphism implies 11

Lemma 2.4. The group Gτ is reductive. In fact, it is isomorphic to


i=1 GLei (C).

Up to conjugation in G(α), it is independent of the

lift x. We denote the conjugacy class of a closed subgroup H of G(α) by (H). Given a reductive subgroup H of G(α), let Mλ (α, θ)(H) denote the set of points x such that the stabilizer of x belongs to (H). We order the conjugacy classes of reductive subgroups of G(α) by (H) ≤ (L) if and only

if L is conjugate to a subgroup of H. The following result is well-known; see [32, Section 4.5] and the references therein. Proposition 2.5. The strata Mλ (α, θ)τ = Mλ (α, θ)(Gτ ) define a finite stratification of Mλ (α, θ) into locally closed subsets such that Mλ (α, θ)(H) ⊂ Mλ (α, θ)(L)

(H) ≤ (L).

Moreover, each stratum is a symplectic leaf with respect to the natural Poisson bracket on Mλ (α, θ). 3. Canonical Decompositions of the Quiver Variety In this section we recall the canonical decomposition of quiver varieties described in [7], and show that it holds in slightly greater generality than stated in loc. cit. ´ 3.1. Etale local structure. In this section, we recall the ´etale local structure of Mλ (α, θ), as described in [8, Section 4]. Since it is assumed in loc. cit. that θ = 0, we provide some details to ensure the results are still applicable in this more general setting. Let x ∈ Mλ (α, θ) be a geometric

point and denote by the same symbol a point in the unique closed G(α)-orbit in µ−1 (λ)θ that maps to x. Then x decomposes into a direct sum xe11 ⊕ · · · ⊕ xekk of θ-stable representations, with

multiplicity. Let β (i) = dim xi . Let Q′ be the quiver with k vertices whose double has 2p(β (i) ) loops at vertex i and −(β (i) , β (j) ) arrows between vertex i and j. The stabilizer of x in G(α) is denoted Gx . The k-tuple e = (e1 , . . . , ek ) defines a dimension vector for the quiver Q′ .

Lemma 3.1. The closed subgroup Gx of G(α) is isomorphic to G(e) and θ|Gx is the trivial character. Proof. The isomorphism Gx ≃ G(e) follows from the fact that HomΠλ (Q) (M1 , M2 ) = 0 if M1 and M2

are non-isomorphic θ-stable representations and EndΠλ (Q) (Mi , Mi ) = C. Under this identification, ′

θ|G(e) = (θ · β (1) , . . . , θ · β (k) ) = (0, . . . , 0) ∈ QQ0 is the trivial stability condition.

If X and Y are two G-varieties, then we say that there is a G-equivariant ´etale isomorphism between a neighborhood of x ∈ X and y ∈ Y if there exists a G-variety Z and equivariant morphisms ψ


Y ←− Z −→ X and z ∈ Z such that φ(z) = x, ψ(z) = y and both φ and ψ are ´etale at z. The proof of the following theorem is identical to the proof of [8, Theorem 4.9], it only requires that 12

one check that the arguments of section 4 of loc. cit. are applicable in this slightly more general setting. Theorem 3.2. There is a G(α)-equivariant ´etale isomorphism between a neighborhood of (1, 0) in −1 θ G(α) ×Gx µ−1 etale isomorphism between a neighborhood of Q′ (0) and x ∈ µ (λ) . This induces an ´

0 in µ−1 Q′ (0)//G(e) and the image of x in Mλ (α, θ).

Proof. In the arguments of section 4 of [8], there are two places where one has to be careful since G(α) · x is only assumed to be closed in µ−1 (λ)θ and not necessarily in µ−1 (λ). Firstly, Lemma 2.4

implies that Gx is reductive, therefore [8, Lemma 4.2] is valid in this setting. Let gx ⊂ g(α) be the

Lie algebra of Gx and choose a Gx -stable compliment L to gx in g(α). By [8, Corollary 2.3], we can choose a Gx -stable coisotropic compliment C to the isotropic subspace [g(α), x] of Rep(Q, α).

Define ν : C → L∗ by ν(c)(l) = ω(c, lx) + ω(c, lc) + ω(c, lx) and let µx be the composite of µ with the restriction map g∗ → g∗x .

Then, in our setting the analogue of [8, Lemma 4.4] states that there exists a Gx -stable affine

open neighborhood U of 0 in C such that the assignment c 7→ c + x induces a Gx -equivariant map −1 (0) → µ−1 (λ)θ , and the induced map U ∩ µ−1 x (0) ∩ ν

−1 (U ∩ µ−1 (0))//G(e) −→ µ−1 (λ)θ //G(α) x (0) ∩ ν

is ´etale at 0. It is crucial here that U be affine so that Luna’s Fundamental Lemma is applicable. To show that the statement holds, first choose n ≫ 0 and an nθ-semi-invariant f such that f (x) 6= 0.

Define h : C → C by c 7→ f (c + x). Then we take U to be the affine open subset on which h does not vanish. Since h(0) = f (x), 0 ∈ U . Lemma 3.1 implies that h is Gx -invariant, therefore U is

Gx -stable. With this in mind, the arguments given in the proof of [8, Lemma 4.4] show that the above stated analogue of that result holds. The remainder of the proof of the theorem follows the proof of [8, Theorem 4.9] verbatim.

cλ (α, θ)x of Mλ (α, θ) at x and the completion M c0 (e, 0)0 of M0 (e, 0) By taking the completion M

at 0, the formal analogue of Theorem 3.2 states

cλ (α, θ)x ≃ M c0 (e, 0)0 . Corollary 3.3. There is an isomorphism of formal schemes M

Remark 3.4. An easy calculation shows that p(α) = p(e). It can also be deduced from the fact cλ (α, θ)x = dim M c0 (e, 0)0 . This fact will be useful later. that dim M

Remark 3.5. Using Corollary 3.3, if α = nβ for β indivisible, then at any point of the stratum of

Mλ (α, θ) corresponding to representations of the form Y ⊕n for dim Y = β, the formal neighborhood is isomorphic to the formal neighborhood of the origin of the variety M0 ((n), 0) for the quiver with one vertex and p(β) arrows. Therefore, if (n, p(β)) 6= (2, 2) and p(β) ≥ 2, we conclude from [26, Theorem B] that M0 ((n), 0) does not admit a symplectic resolution, and since it is a cone, the statement is equivalent to the same about the formal neighborhood of zero (cf. Lemma 6.15 13

below). Therefore a formal neighborhood of any point in this stratum does not admit a symplectic resolution, so neither can Mλ (α, θ). This gives an alternative proof of the non-existence portion of Theorem 1.4. 3.2. Hyperk¨ ahler twisting. Let α = m1 ν (1) + · · · + mt ν (t) be the canonical decomposition of α

with respect to Σλ . It is shown in [7] that

Theorem 3.6. There is an isomorphism of varieties



 S mi Mλ (ν (i) , 0) ≃ Mλ (α, 0).

Moreover, if ν (i) is real then S mi (Mν (i) (λ, 0)) = {pt} and if ν (i) is non-isotropic imaginary then

mi = 1.

Theorem 3.7. Let α = n1 σ (1) + · · · + nk σ (k) be the canonical decomposition of α with respect to Σλ,θ . Then, there is an isomorphism of Poisson varieties   Y ∼ φ: S ni Mλ (σ (i) , θ) −→ Mλ (α, θ). i

The proof of Theorem 3.7 is given at the end of section 3.3. In order to deduce Theorem 3.7

from [7, Theorem 1.1], we use hyperk¨ahler twists. By our main assumption (1), λ ∈ RQ0 . Proposition 3.8. Let ν = −λ − iθ and consider Mλ (α, θ), Mν (α, 0) as complex analytic spaces. Hyperk¨ ahler twisting defines a homeomorphism of stratified spaces ∼

Ψ : Mλ (α, θ) −→ Mν (α, 0), ∼

i.e. Ψ restricts to a homeomorphism Mλ (α, θ)(H) −→ Mν (α, 0)(H) for all classes (H). In particular,

the homeomorphism maps stable representations to stable (= simple) representations.

Proof. We follow the setup described in the proof of [5, Lemma 3]. We have moment maps √ X −1 X [xa , xa∗ ], µR (x) = µC (x) = [xa , x†a ] + [xa∗ , x†a∗ ]. 2 a∈Q1


As shown in [27, Corollary 6.2], the Kempf-Ness Theorem says that the embedding µ−1 C (λ) ∩

−1 µ−1 R (iθ) ֒→ µC (λ) induces a bijection

−1 µ−1 C (λ) ∩ µR (iθ)/U (α) −→ Mλ (α, θ).


−1 Since the embedding is clearly continuous and the topology on the quotients µ−1 C (λ)∩ µR (iθ)/U (α)

and Mλ (α, θ) is the quotient topology (for the latter space, see [40, Corollary 1.6 and Remark 1.7]), the bijection (3) is continuous. −1 Define a stratification µ−1 C (λ)∩µR (iθ)/U (α) analogous to the stratification of Mλ (α, θ) described −1 in section 2.5. Let x ∈ Mλ (α, θ) have a θ-polystable lift x = xe11 ⊕ · · · ⊕ xekk in µ−1 C (λ) ∩ µR (iθ).

Then Lemma 3.1 says that Gx = G(e) and [27, Proposition 6.5] implies that U (α)x = U (e). Hence

G(α)x = U (α)C x . Therefore the homeomorphism (3) restricts to a bijection −1 (µ−1 C (λ) ∩ µR (iθ)/U (α))(K) → Mλ (α, θ)(K C ) 14

for each (K). Let the quaternions H = R ⊕ Ri ⊕ Rj ⊕ Rk act on Rep(Q, α) by extending the usual complex

structure so that j · (xa , xa∗ ) = (−x†a∗ , x†a ). In general,

(z1 + z2 j) · (xa , xa∗ ) = (z1 xa − z2 x†a∗ , z1 xa∗ + z2 x†a ). This action commutes with the action of U (α) and satisfies µR (z · x) = (||z1 ||2 − ||z2 ||2 )µR (x) − iz1 z 2 µC (x) − iz2 z 1 µC (x)† ,


µC (z · x) = z12 µC (x) − z22 µC (x)† − 2iz1 z2 µR (x),


∀ z ∈ H.

√ Let h = (i − j)/ 2. Then multiplication by h defines a homeomorphism ∼

−1 −1 −1 µ−1 C (λ) ∩ µR (iθ) −→ µC (−λ − iθ) ∩ µR (0)

Since multiplication by h commutes with the action of U (α), this homeomorphism descends to a homomorphism   ∼ −1 −1 −1 µ−1 C (λ) ∩ µR (iθ) /U (α) −→ µC (−λ − iθ) ∩ µR (0) /U (α)

which preserves the stratification by stabilizer type.

Thus, the map Ψ is the composition of three homeomorphisms, each of which preserves the stratification.

Remark 3.9. Our general assumption that λ ∈ RQ0 if θ 6= 0 is required in the proof of Proposition

3.8 to ensure that multiplication by h lands in µ−1 R (0). Equation (4) implies that it would suffice to assume more generally that there exists z ∈ C such that |z| = 1 and zλ ∈ RQ0 . It is natural to

expect that Theorem 3.7 holds with out the assumption λ ∈ RQ0 .

Remark 3.10. Using the notion of smooth structures on stratified symplectic spaces, as defined in [41], one can presumably strengthen Proposition 3.8 to the statement that there is a diffeomorphism ∼

of stratified symplectic spaces Mλ (α, θ) −→ Mν (α, 0). Proposition 3.11. The variety Mλ (α, θ) is irreducible and normal. Proof. We begin by showing that the variety Mλ (α, θ) is connected. Proposition 3.8 implies that Mλ (α, θ) is connected if and only if Mν (α, 0) is connected. The latter is known to be connected by [7, Corollary 1.4]. Next, we show that Mλ (α, θ) is irreducible. Since Mλ (α, θ) is connected, it suffices to show that, for each C-point x ∈ Mλ (α, θ), the local ring OMλ (α,θ),x is a domain. This ring embeds into the

formal neighborhood of x in Mλ (α, θ). By Corollary 3.3, the formal neighborhood of x in Mλ (α, θ)

is isomorphic to the formal neighborhood of 0 in M0 (e, 0). By [7, Corollary 1.4], this is a domain. Finally, normality is an etal´e local property, [33, Remark 2.24 and Proposition 3.17]. Therefore, as in the previous paragraph this follows from Theorem 3.2 and [8, Theorem 1.1]. 15

3.3. The proof of Theorem 3.7. Recall that α = n1 σ (1) + · · · + nk σ (k) is the canonical decom-

+ position of α in Rλ,θ . The map φ is defined as follows. Let H(α) be the product G(σ (1) )n1 ×

· · · × G(σ (k) )nk , thought of as a subgroup of G(α). There is a natural H(α)-equivariant inclusion Q ∗ (i) ni ֒→ T ∗ Rep(Q, α). This is an inclusion of symplectic vector spaces. Since the i T Rep(Q, σ )

moment map for the action of H(α) on T ∗ Rep(Q, α) is the composition of the moment map for G(α) followed by projection from the Lie algebra of G(α) to the Lie algebra of H(α), the above Q θ inclusion restricts to an inclusion i (µ−1 (λ)θ )ni ֒→ µ−1 α (λ) , inducing a map of GIT quotients σ(i) Y Mλ (σ (i) , θ)ni → Mλ (α, θ). i

This map, which sends a tuple of representations (Mi,j ) to the direct sum  Q through i S ni Mλ (σ (i) , θ) . It is this map that we call φ.



Mi,j clearly factors

Passing to the analytic topology, Proposition 3.8 implies that we get a commutative diagram Q Q



 Mλ (σ (i) , θ) 


S ni

(−σ (i)

/ Mλ (α, θ)


 / M−λ−iθ (α, 0).

 − iθ, 0)

where both vertical arrows are homeomorphisms and the bottom horizontal arrow is an isomorphism by Theorem 3.6. Therefore, we conclude that φ is bijective. Since we are working over the complex numbers, and we have shown in Proposition 3.11 that Mλ (α, θ) is normal, we conclude by Zariski’s main theorem that φ is an isomorphism. Corollary 3.12. The variety Mλ (α, θ) has dimension 2


i=1 ni p(σ

(i) ).

Proof. By Theorem 3.7, it suffices to show that dim Mλ (α, θ) = 2p(α) if α ∈ Σλ,θ . We note that

Proposition 3.8, together with the results of [6], imply that there exists a θ-stable representation of Πλ (Q) of dimension α if and only if α ∈ Σλ,θ . Let U be the subset of Mλ (α, θ) consisting of θ-stable

representations. Since α is assumed to be in Σλ,θ , Proposition 3.11 implies that U is a dense open subset of Mλ (α, θ). Let V be the open subset of Rep(Q, α) consisting of θ-stable representations.

Then U is the image of µ−1 (λ) ∩ V under the quotient map and hence V is non-empty. The group

G(α)/C× acts freely on V and µ is smooth when restricted to V . Thus,

dim U = dim Rep(Q, α) − 2(dim G(α) − 1) = 2p(α), as required.

Finally, we need to check that the morphism φ is Poisson. Since both varieties are normal by Proposition 3.11, it suffices to show that φ induces an isomorphism of symplectic manifolds  Q between the open leaf of Mλ (α, θ) and the open leaf of i S ni Mλ (σ (i) , θ) . By Proposition 2.5, the symplectic leaves of Mλ (α, θ) are the strata given by stabilizer type. Therefore the explicit 16

description of φ given at the start of this section shows that φ restricts to an isomorphism between strata. In particular, φ restricts to an isomorphism between the open leaves. The symplectic structure on the open leaf of Mλ (α, θ) comes from the symplectic structure on T ∗ Rep(Q, α). More specifically, the non-degenerate closed form on the latter space restricts to a degenerate G(α)-equivariant two-form on µ−1 (λ)θ . Hence it descends to a closed two-form on Mλ (α, θ). The restriction of this two-form to the open leaf is non-degenerate. The two-form on the  Q open leaf of i S ni Mλ (σ (i) , θ) is defined similarly. Now the point is that under the embedding Q −1 Q −1 θ θ ni is simply the θ ni −1 i (µσ(i) (λ) ) i (µσ(i) (λ) ) ֒→ µα (λ) , the H(α)-equivariant closed two-form on θ pull-back of the G(α)-equivariant closed two-form on µ−1 α (λ) . This implies that the two-form on  Q ni the open leaf of i S Mλ (σ (i) , θ) is the pull-back, under φ, of the symplectic two-form on the

open leaf of Mλ (α, θ).

4. Smooth v.s. stable points As usual, choose deformation parameter λ ∈ RQ0 , stability parameter θ ∈ QQ0 and dimension

+ The main goal of this section is to prove Theorem 1.13, which says that x ∈ vector α ∈ NRλ,θ

Mλ (α, θ) is θ-canonically stable if and only if it is in the smooth locus of Mλ (α, θ).

4.1. The proof of Theorem 1.13. The proof of Theorem 1.13 follows closely the arguments given in [29, Theorem 3.2]. We provide the necessary details that show that the arguments of loc. cit. are valid in our setting. First, notice that, under the isomorphism of Theorem 3.7 (2), the open subset of θ-canonically stable points in Mλ (α, θ) is the product of the θ-canonically stable points in the spaces S ni Mλ (σ (i) , θ). Therefore it suffices to show that the set of θ-canonically stable points in S ni Mλ (σ (i) , θ) is precisely the smooth locus. If σ (i) is real then S ni Mλ (σ (i) , θ) is a point. If σ (i) is an isotropic imaginary root then Mλ (σ (i) , θ) is a partial resolution of a Kleinian singularity. In particular, it is a 2-dimensional quasi-projective variety. This implies that the smooth locus of S ni Mλ (σ (i) , θ) equals S ni ,◦ Mλ (σ (i) , θ)sm :=

 ni X 


  pj pj ∈ Mλ (σ (i) , θ)sm , pj 6= pk for j 6= k . 

On the other hand, the set of θ-canonically stable points in S ni Mλ (σ (i) , θ) equals S ni ,◦ U , where U ⊂ Mλ (σ (i) , θ) is the set of θ-canonically stable points. Therefore, in this case it suffices to show that Mλ (σ (i) , θ)sm equals U . Finally, in the case where σ (i) is a non-isotropic imaginary root,

ni = 1. Thus, we are reduced to considering the situation where α ∈ Σλ,θ is an imaginary root. In this

case, a point x is θ-canonically stable if and only if it is θ-stable. As in the proof of Corollary 3.12, it is clear from the definition of Mλ (α, θ) that the set of θ-stable points is contained in the smooth locus. Therefore it suffices to show that if x is not θ-stable then it is a singular point. As in section 3.1, decompose x into a direct sum xe11 ⊕ · · · ⊕ xeℓ ℓ of θ-stable representations with multiplicity. Let 17

β (i) = dim xi . Let Q′ be the quiver with ℓ vertices whose double has 2p(β (i) ) loops at vertex i and −(β (i) , β (j) ) arrows between vertex i and j. The ℓ-tuple e = (e1 , . . . , eℓ ) defines a dimension vector

for the quiver Q′ . By Theorem 3.2, it suffices to show that 0 is contained in the singular locus of M0 (e, 0). In order to proceed, we require [28, Proposition 1.1], stated in our generality. The proof is identical to the proof given in loc. cit. this time using Theorem 3.2. Proposition 4.1. Assume that α ∈ Σλ,θ and let x be a geometric point of Mλ (α, θ), of represen-

tation type τ = (e1 , β1 ; . . . ; ek , βk ). Then e is the dimension vector of a simple Π0 (Q′ )-module i.e.

e ∈ Σ0 (Q′ ). Returning to the proof of Theorem 1.13, with Proposition 4.1 in hand, the argument given in the proof of [29, Theorem 3.2] goes through verbatim. This completes the proof of Theorem 1.13. 4.2. The proof of Corollary 1.15. By Theorem 1.13, Mλ (α, θ) is smooth if and only if every point is θ-canonically stable. As in the reduction argument given at the start of the proof of Theorem 1.13, this means that ni must be 1 when σ (i) is an isotropic imaginary root. Moreover, it is clear that Mλ (σ (i) , θ) consists only of θ-stable points if and only if σ (i) is minimal. 5. The variety X(n, d) Recall that X(n, d) denotes the quiver variety ) d X n [Xi , Yi ] = 0 // GL(n, C). (X1 , Y1 , . . . , Xd , Yd ) ∈ EndC (C )



In this section we recall results of Lehn-Kaledin [25] and Lehn-Kaledin-Sorger [26], which say when X(n, d) admits a projective symplectic resolution. We note that X(n, d) is an irreducible, normal affine variety of dimension 2(n2 (d − 1) + 1). 5.1. The case (n, d) = (2, 2). Let W = sl2 and (V, ω) a 4-dimensional symplectic vector space. Let κ denote the Killing form on W . Then κ ⊗ ω is a symplectic form on W ⊗ V . We identify

sp(V )∗ with sp(V ) via its Killing form. There is an action of PGL(2) on W by conjugation and hence on W ⊗ V . This action is Hamiltonian and commutes with the natural action of Sp(V ) on

W ⊗ V . The moment map for the action of PGL(2) is given by ! X X Ai Aj ω(vi , vj ) Ai ⊗ vi = µ i,j



X i

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